We show that one can obtain analytic solutions of the time-dependent Schrödinger equation that are more complex than the well-known oscillating coherent wave packet. Such Hermite-Gaussian or initially square wave packets exist for a free particle or for one subject to the harmonic oscillator potential. In either case, the Hermite-Gaussian packets retain their nodal structure even after long times. There is a single class of exact solutions for the system with oscillator constant K > 0, K = 0, or K < 0, leading to wave functions for the harmonic oscillator, the free particle, and the inverted oscillator, respectively.

1.
E.
Schrödinger
, “
Der stetige Übergang von der Mikro-zur Makromechanik
,”
Naturwissenschaften
14
,
664
666
(
1926
).
2.
Leonard I.
Schiff
,
Quantum Mechanics
, 3rd ed. (
McGraw-Hill
,
New York
,
1968
), pp.
74
76
.
3.
Alexander N.
Drozdov
, “
Power-series expansion for the time evolution operator with a harmonic oscillator reference system
,”
Phys. Rev. Lett.
75
,
4342
4345
(
1995
).
4.
W.
van Dijk
and
F. M.
Toyama
, “
Accurate numerical solutions of the time-dependent Schrödinger equation
,”
Phys. Rev. E
75
,
036707
(
2007
).
5.
Hezhu
Shao
and
Zhongcheng
Wang
, “
Numerical solutions of the time-dependent Schrödinger equation: Reduction of the error due to space discretization
,”
Phys. Rev. E
79
,
056705
(
2009
).
6.
W.
van Dijk
,
J.
Brown
, and
K.
Spyksma
, “
Efficiency and accuracy of numerical solutions to the time-dependent Schrödinger equation
,”
Phys. Rev. E
84
,
056703
(
2011
).
7.
J. J.
Sakurai
,
Modern Quantum Mechanics
, revised ed. (
Addison-Wesley
,
Reading, MA
,
1994
), p. 112.
8.
F. A.
Barone
,
H.
Boschi-Filho
, and
C.
Farina
, “
Three methods for calculating the Feynman propagator
,”
Am. J. Phys.
71
,
483
491
(
2003
).
9.
Kiyoto
Hira
, “
Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations
,”
Eur. J. Phys.
34
,
777
785
(
2013
).
10.
The phase of the propagator (and wave function) is a continuous and unambiguous function of t if one uses the Feynman-Soriau formula, which is Eq. (4) with the factor multiplying the exponential replaced by eiπ2[ωt/π]α2/(2πi|sinωt|), where [ωt/π] is the greatest integer smaller than ωt/π. See
Bjorn
Felsager
,
Geometry, Particles, and Fields
(
Odense U.P.
,
Denmark
,
1981
), pp.
189
191
.
11.
Actually, the domain of functions that properly represents initial states is more limited than that of square-integrable functions, even if the functions and all their derivatives in x are continuous. For discussions of a counter-example see
Barry R.
Holstein
, “
Spreading wave packets—a cautionary note
,”
Am. J. Phys.
40
,
829
832
(
1972
) and
James R.
Klein
, “
Do free quantum-mechanical wave packets always spread?
,”
Am. J. Phys.
48
,
1035
1037
(
1980
).
12.
Izuru
Fujiwara
and
Kunio
Miyoshi
, “
Pulsating states for quantal harmonic oscillator
,”
Prog. Theor. Phys.
64
,
715
718
(
1980
).
13.
A. S.
de Castro
and
N. C.
da Cruz
, “
A pulsating Gaussian wave packet
,”
Eur. J. Phys.
20
,
L19
L20
(
1999
).
14.
S.
Waldenstrøm
and
K.
Razi Naqvi
, “
A neglected aspect of the pulsating gaussian wave packet
,”
Eur. J. Phys.
20
,
L41
L43
(
1999
).
15.
M. E.
Marhic
, “
Oscillating Hermite-Gaussian wave functions of the harmonic oscillator
,”
Lett. Nuovo Cim.
22
,
376
378
(
1978
).
16.
S. I.
Gradshteyn
and
I. M.
Ryzhik
,
Table of Integrals, Series, and Products
(
Academic Press, Inc.
,
San Diego
,
1980
), Eq. (7.374.8).
17.
Guang-Jie
Guo
,
Zhong-Zhou
Ren
,
Guo-Xing
Ju
, and
Chao-Yun
Long
, “
The sojourn time of the inverted harmonic oscillator on the noncommutative plane
,”
J. Phys. A: Math. Theor.
44
,
425301
(
2011
).
18.
Guang-Jie
Guo
,
Zhong-Zhou
Ren
,
Guo-Xing
Ju
, and
Xiao-Yong
Guo
, “
Quantum tunneling effect of driven inverted harmonic oscillator
,”
J. Phys. A: Math. Theor.
44
,
305301
(
2011
).
19.
Guang-Jie
Guo
,
Zhong-Zhou
Ren
,
Guo-Xing
Ju
, and
Xiao-Yong
Guo
, “
Quantum tunneling effect of a time-dependent inverted harmonic oscillator
,”
J. Phys. A: Math. Theor.
44
,
185301
(
2011
).
20.
H. O.
Girotti
, “
Noncommutative quantum mechanics
,”
Am. J. Phys.
72
,
608
612
(
2004
).
21.
Anirban
Saha
and
Sunandan
Gangopadhyay
, “
Noncommutative quantum mechanics of a harmonic oscillator under linearized gravitational waves
,”
Phys. Rev. D
83
,
025004
(
2011
).
22.
Mark
Andrews
, “
The evolution of free wave packets
,”
Am. J. Phys.
76
,
1102
1107
(
2008
).
23.
T.
Hannesson
and
S. M.
Blinder
, “
Theta-function representation for the particle-in-a-box propagator
,”
Il Nuovo Cim. B
79
,
284
290
(
1984
).
24.
S. A.
Fulling
and
K. S.
Güntürk
, “
Exploring the propagator of a particle in a box
,”
Am. J. Phys.
71
,
55
63
(
2003
).
25.
R. W.
Robinett
, “
Quantum wave packet revivals
,”
Phys. Rep.
392
,
1
119
(
2004
).
AAPT members receive access to the American Journal of Physics and The Physics Teacher as a member benefit. To learn more about this member benefit and becoming an AAPT member, visit the Joining AAPT page.